A significant body of literature has addressed the strategic design of biomass supply chain networks. Abandansari et al.
Another stream of research has focused on uncertainty, resilience, and risk management in biomass supply chains. Allegranzi et al.
Approach examined in the present article.
This figure is shown in the PDF version of the article.
In biomass energy supply chain network design, one of the fundamental steps is the effective aggregation of spatially dispersed agricultural areas to define the network's input nodes. Since farm-related spatial data are typically characterized by high dispersion, irregular geographical boundaries, and heterogeneous production capacities, clustering techniques serve as an essential preprocessing tool for reducing problem complexity and improving network coherence. Accordingly, four clustering algorithms, CKM, GMM, AP, and SC, are employed in this study to identify optimal spatial groupings of farms and to facilitate a comprehensive evaluation in terms of clustering quality, operational feasibility, and transportation efficiency.
The first method, CKM, extends the traditional K-Means algorithm by incorporating additional constraints on cluster size and composition. The algorithm seeks to allocate observations to (K) clusters in such a way that the total within-cluster sum of squared distances is minimized while satisfying predefined constraints. The optimization problem is expressed in Eq. (1).
min∑ ∑ ‖xj − μi‖ subject to constraint on |Ci|, (1)
i=1 xi∈Ci
where xi denotes the input data points, μi represents the centroid of cluster i, and Ci is the set of observations assigned to cluster i. This approach is particularly useful when collection-center capacities are limited and explicit control over cluster sizes is required.
In contrast, the GMM adopts a probabilistic perspective by representing the data as a weighted combination of multivariate Gaussian distributions. The probability density function is defined as
p(x) = ∑ πiΝ(x|μi, ∑ ), (2)
i=1 i
where πi, and ∑i denote the mixture weight, mean vector, and covariance matrix of component i, respectively. Parameter estimation is performed using the Expectation-Maximization (EM) algorithm, which iteratively maximizes the log-likelihood function:
max log ∏ ∑ πiΝ(x|μi, ∑ ). (3)
i=1 i
Compared with K-Means, GMM provides greater flexibility in modeling overlapping clusters and datasets with ambiguous cluster boundaries.
The third technique, AP, differs from conventional clustering methods in that it does not require the number of clusters to be specified a priori. Instead, clusters are formed through an iterative message-passing process between data points. Two types of messages are exchanged: responsibility, which reflects the suitability of one point to serve as the exemplar of another, and availability, which indicates the degree to which a point is willing to act as an exemplar. The clustering objective is to maximize the total similarity between observations and their selected exemplars:
max ∑ s(i, ci), (4)
where s(i, ci) denotes the similarity between point i and its assigned exemplar ci. Through this iterative process, the algorithm is also capable of automatically determining the appropriate number of clusters. The fourth approach, SC, utilizes spectral graph theory to identify complex and nonlinear structures within the data. Initially, a similarity matrix W is constructed among all observations. Subsequently, the normalized graph Laplacian is computed as
Lsym = I − D−2WD−2. (5)
where D is the diagonal degree matrix. After computing the eigenvectors of Lsym, the data are projected into a lower-dimensional space and clustered using K-Means. SC is particularly effective for identifying complex and non-convex cluster structures. In this study, the four clustering algorithms are employed to generate alternative spatial configurations of biomass resources rather than merely compare algorithm performance. The resulting diversity in cluster structures enables a comprehensive evaluation of transportation efficiency and network performance, providing a robust basis for selecting the most suitable configuration for biomass supply chain network design.
To identify the most suitable locations for biomass hubs, a two-stage MCDM framework is employed. First, criterion weights are determined using the B-BWM, which accounts for uncertainty in decision-makers’ judgments by modeling preferences as probability distributions rather than fixed values. This approach provides more reliable and robust criterion weights while enabling improved sensitivity analysis.
Subsequently, candidate hub locations are evaluated and ranked using the Best-Worst PROMETHEE method. By combining the robust weighting capability of B-BWM with the ranking effectiveness of PROMETHEE, this hybrid approach improves decision transparency, reduces subjective bias, and enhances the reliability of the final ranking results. The obtained hub scores are then incorporated into the optimization model as an input for hub location decisions. Due to space limitations, the detailed computational procedures are omitted, and interested readers are referred to
In this section, the mathematical model for biomass energy network design is presented within a two-phase framework: the first phase is dedicated to hub location and cluster allocation, while the second phase models the complete network structure by accounting for product and energy flows and uncertainties. The complete list of symbols and indices is provided in
Notation and Indexing.
This table is shown in the PDF version of the article.
us it Visit the position of land i in period t under scenario s xijt s 1 if route goes from farm i to farm j in period t under scenario s; 0 otherwise.
Phase 1 Parameters πh Score of hub h disηh Distance between cluster η and hub h K Maximum number of hubs to be established.
η Maximum number of clusters that can be connected to one hub.
Phase 1 Decision Variables ωht 1 if hub h is established in period t; 0 otherwise.
Zη t h 1 if cluster η is assigned to hub h in period t; 0 otherwise.
MaxZ1 = ∑ πh × ωht. (6)
MinZ2 = ∑ dis Zt . (7)
ηh ηh η,H,T
t (8)
∑ Zηh = 1, ∀η, T.
t (9)
∑ Zηh ≤ η , ∀H, T.
(10)
∑ ωht ≤ K, ∀T.
Zη t h ≤ ωηt, ∀η, H, T. (11) Zη t h, ωηt ∈ {0,1}. (12)
Eq. (6) maximizes total hub scores across planning periods, while Eq. (7) minimizes total distance between clusters and their assigned hubs. Constraints (8)-(11) ensure each cluster is assigned to exactly one active hub per period, limit cluster-to-hub assignments, and control the number of active hubs. The bi-objective model is solved using the weighted-sum method.
s s s
MinZ′ = ∑ PSs ( ∑ Ctphr × Pphrt + ∑ Ctrc × Urct + ∑ Ctcc′ × Ucc′t
S P,H,R,T R,C,T C,C′,T
+ ∑ Ctc′t′ × Uc′t′t + ∑ Ctt′d × ELt′dt + α ( ∑ Fe × yet)
C′,T′,T T′,D,T T,e∈{h,r,c,t′}
s s main main (13)
+ α ( ∑ φrct × Cprc + ∑ φcc′t × Cpcc′) + ∑ hcts × ϕphts
R,C,T C,C′,T P,H,T adj adj w s I s
+ ∑ hcts × ϕphts + ∑ hcts × Wpht + ∑ hcts × Ipht
P,H,T P,H,T P,H,T
+ ∑ xijt s × disijη × times ijηt − ∑ ELs t′dt × priced),
i,j,T,η T′,D,T
η (14)
main adj ′ s
ϕphts + ϕphts = ∑ Zηht × ∑ qpft , ∀H, P, T, S,
η=1 f∈Fη
∑ ϕmain ≤ Cap + M × y , ∀H, T, S, (15)
phts h ht ∑ ϕadj ≤ Cap × y , ∀H, T, S, (16)
phts h ht
∑ ϕadj ≥ ∑ ϕmain − Cap , ∀H, T, S, (17)
phts phts h
I + ϕmain + ϕadj = ∑ PS + I + w , ∀P, H, T, S, (18)
ph(t−1)s phts phts phrt phts phts
Pphrt S ≤ M × φs hrt, ∀H, R, P, T, S, (19) S (20)
∑ Pphrt ≤ capr, ∀R, T, S,
∑ PS (1 − α) ≤ ∑ Us , ∀P, R, T, , S, (21)
phrt rct
∑ Us (1 − α) ≤ ∑ Us ′ , ∀C, T, S, (22)
rct cc t R C′
∑ Us ′ (1 − α) ≤ ∑ Us′ ′ ∀C, T′, T, S, (23)
cc t c t t C′ C′
∑ Us′ ′ (1 − α) ≤ ∑ ELs′ , ∀T′, T, S, (24)
c t t t dt C′ D
Urct s ≤ M × EPrc S × φs rct, ∀R, C, T, S, (25) Uc sc′t ≤ M × yct, ∀C, T′, T, S, (26) Uc sc′t ≤ M × EPcSc′ × φs cc′t, ∀C, T′, T, S, (27) Urct s ≤ M × yrt, ∀R, C, T, S, (28) ELt′dt ≤ M × yt′t, ∀T , D, T, S, (29)
∑ ELs′ ≤ I ˜ × M, ∀D, T, S, (30)
t dt dt
us + us ≥ ∑ q + Q(1 − xs ), ∀(i. j) ∈ F , η, T, S , i ≠ j, (31)
it jt pjts ijt η
∑ xs = ∑ xs , ∀j ∈ F , η, T, S, (32)
ijt jit η i∈Fη i∈Fη
∑ xs = 1, ∀j ∈ F , η, T, S, (33)
ijt η
∑ xs = 1, ∀η, T, S, (34)
1jt j∈Fη
∑ qs ≤ us , ∀η, i ∈ F , T, S, (35)
pit it η
∑ us ≤ Q, ∀η, T, S, (36)
it i∈Fη
Pphrt s , Urct s , Us ′ , Uc′t′t s , ELs t′dt , ϕmain phts , ϕadj , Ipht s , Wpht s ≥ 0, (37)
cc t phts
yet , φs rct, φs cc′t, xijt s , ωht , Zη t h ∈ {0.1}. (38)
Eq. (13) minimizes total supply chain costs, including transportation, facility establishment, inventory holding, and intra-cluster routing, net of electricity revenue. Constraints (14)-(18) govern biomass flow from farms to hubs, warehouse activation priorities, and inventory balance. Constraints (19)-(24) regulate biomass-to-energy conversion flows across reactors, condensers, and transformers. Constraints (25)-(30) ensure transfers occur only through active facilities and links, with electricity supplied only to genuine demand regions. Finally, Constraints (31)-(36) manage intra-cluster vehicle routing by eliminating sub-tours, enforcing farm visits, ensuring vehicle return, and imposing capacity limits. Constraints (37)-(38) define variable domains.